To determine the particle's velocity, we can use the concept of relativistic mass and the equation relating relativistic mass to rest mass and velocity.
The equation for relativistic mass is given by:
m_rel = γ * m_0,
where: m_rel is the relativistic mass, m_0 is the rest mass, γ is the Lorentz factor.
In this case, we are given that the relativistic mass (m_rel) is 1.25 times bigger than the rest mass (m_0). Therefore, we can write:
m_rel = 1.25 * m_0.
Substituting this into the equation for relativistic mass, we have:
1.25 * m_0 = γ * m_0.
Canceling out the m_0 term, we get:
1.25 = γ.
The Lorentz factor (γ) is given by:
γ = 1 / sqrt(1 - v^2/c^2),
where: v is the velocity of the particle, c is the speed of light in a vacuum (approximately 299,792,458 meters per second).
Substituting the value of γ into the equation, we have:
1.25 = 1 / sqrt(1 - v^2/c^2).
Squaring both sides of the equation, we get:
1.5625 = 1 / (1 - v^2/c^2).
Inverting both sides of the equation, we have:
1/(1.5625) = 1 - v^2/c^2.
Simplifying, we get:
0.64 = 1 - v^2/c^2.
Rearranging the equation, we have:
v^2/c^2 = 1 - 0.64.
v^2/c^2 = 0.36.
Taking the square root of both sides, we get:
v/c = sqrt(0.36).
v/c = 0.6.
Finally, multiplying both sides by c, we have:
v = 0.6 * c.
Therefore, the particle's velocity is 0.6 times the speed of light.
Note: The result is expressed as a fraction of the speed of light because it is a relativistic calculation, and velocities close to the speed of light are best represented in this manner.